Image Cover Sheet CLASSIFICATION SYSTEM NUMBER 52503 … · of uniformly stiffened cylinders subject to hydrostatic pressure have been assembled and incorporated in the comput,er

Image Cover Sheet

CLASSIFICATION SYSTEM NUMBER 52503

UNCLASSIFIED lllllllllllllllllllllllllllllllllll TITLE

PRHDEF - STRESS AND STABILITY ANALYSIS OF RING STIFFENED SUBMARINE PRESSURE HULLS

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National Defence Defense nationale Research and Bureau de recherche Development Branch et developpement

TECHNICAL MEMORANDUM 87/213 June 1987

PRHDEF -STRESS AND STABILITY ANALYSIS

OF RING STIFFENED SUBMARINE PRESSURE HULLS

N.G. Pegg

Defence Research Establishment Atlantic

Canada

D.R. Smith

Centre de Recherches pour Ia Defense Atlantique

DEFENCE RESEARCH ESTABLISHMENT ATLANTIC g GROVE STREET P.O. BOX 1012

DARTMOUTH, N.S,

B2Y 3 Z7 TE:LEPHONE

(9021 426-3100

CENTRE DE RECHERCHES POUR LA DEFENSE ATLANTIQUI 9 GROVE STREET C.P. 1012

DARTMOUTH, N.E.

B2Y 3Z7

I+ UNLIMITED DISTRIBUTION

National Defence Defense nationale Research and Bureau de recherche Development Branch et developpement

PRHDEF -STRESS AND STABILITY ANALYSIS

OF RING STIFFENED SUBMARINE PRESSURE HULLS

N.G. Pegg D.R. Smith

June 1987

Approved by B.F. Peters A/Director/Technology Division

DISTRIBUTION APPROVED BY g1\k_ A/0/TD

TECHNICAL MEMORANDUM 87/213


Centre de Recherches pour Ia Defense

Canada i

Abstract

/preliminary investigation of the structural integrity of a submarine pressure hull can be accomplished by the use of design formulae. Approximate solutions for stress and stability of uniformly stiffened cylinders subject to hydrostatic pressure have been assembled and incorporated in the comput,er code PRHDEF. The British pressure vessel code, BS5500, and other codes have been used where appropriate. Critical pressures are determined for yielding in the frames and shell, for interframe and overall bifurcation buckling and for collapse of the stiffened shell and endcap. The effect of out of circularity on frame failure is considered and dimension checks for stiffener tripping are made. The background and limitations of the various equations are discussed and results are compared with those obtained using the axisymmetric finite difference program BOSOR4~

The methods described in this report are particularly useful for comparison of various design alternatives on a common basis and for preliminary investigation before more complex and costly finite element or finite difference analyses are undertaken.

'

Resume

L'etude preliminaire de la solidite structurale de la coque epaisse d'un sous-marin peut etre effectuee a !'aide de formules theoriques. Des solutions approximatives au probleme des contraintes et de la stabilite de cylindres renforces soumis a la pression hydrostatique ont ete assemblees et incorporees au code machine PRHDEF. Le code anglais pour vaisseaux sous pression BS5500 et d'autres codes ont <~te utilises lorsque appropries. Les pressions critiques sont determit1ees pour ce qui est de la deformation des couples et de la coque, du flambage tie bifurcation inter-couples glmerale ainsi que de 1' affaissement de la cc,que et du capot renforces. On prend en consideration .l'effet de la non circularite sur la defaillance des couples et on effectue des verifications des ciimensions pour mesurer le flambage des couples. On examine l'historique ei~ les limites des diverses equations et les resultats sont compares a ceux obtenus au moyen du programme BOSOR4 aux differences finies axi-symetriques.

Les methodes decrites dans le present rapport sont particulierement utiles pour comparer diverses solutions theoriques a partir d'une base commune et pour faire une etude pt~eliminaire avant d'entreprendre des analyses aux elements finis ou aux clifferences finies plus complexes et plus couteuses.

ii

Contents

Abstract ii

Table of Contents iii

List of Figures iv

List of Tables iv

Notation v

1 Introduction 1

2 Theory 2 2.1 Description of Submarine Geometry . . . . . . . . . 2 2.2 Determination of Effective Lengths of Shell Plating . 2 2.3 Effects of Residual Stresses . . . . . . . . 3 2.4 Interframe Bifurcation Buckling Pressures 4 2.5 Determination of Stress Values . . . . . 4 2.6 BS5500 Collapse Curve for Cylinders . . 6 2. 7 Overall Buckling Pressure . . . . . . . . 7 2.8 Out of Circularity and Frame Collapse . 8 2.9 Stiffener Buckling . 9 2.10 Endcap Collapse . . . . . . . . . . . . . 9

3 Discussion 10

4 Conclusions 11

Appendix A: Sample Terminal Input Session of PRHDEF 20

Appendix B: Sample Output Analysis of PRHDEF for Three Models 24:

• References 35

iii

List of Figures

1 2 3 4 5 6

Geometry of Cylinder and Endcap Section . Effective Length Tables2 • • • • •

Interframe Buckling; Mode2 • • •

Collapse Pressure Design Curve3

Overall Buckling Mode Shape . . Stress vs Pressure Ftelationship1

List of Tables

Comparison of Effective Length Formulae . . . . . Comparison of Interframe Buckling Pressures . . . Comparison of Linear and Nonlinear Stress Values Comparison of Overall Buckling Pressures . . . . .

iv

16 17 18 18 19 19

13 14 14 15

Notation

a

A

c

d

D

e

E

I

G

G

h

H

mean radius of cylinder

endcap radius

a2At (f

area of ring stiffener

faying flange width

a{l- Pa.8a2}.1 L 2EhL2 2

cylinder out of circularity, OOC

a{ Pa8a2 }1 L 1 + 2EhL2 2

Eh8

12(1-vli)

distance from cylinder axis to frame centroid

depth of frame web

cL T

distance from shell center to frame centroid

Young's modulus

dL T

Pa2(1-f)T Eh

[ . h"' .!:!+ h"' . "'I -2 810 a cos 2 cos asma sinh a:+sin a

shell thickness

-~(esin/ cos/+ /sinh ecoshe)

{[1+ -2--~~~----~siin~h~a~+~siin~a~~-----------

he thickness of endcap shell

v

hw thickness of frame web

h 1 thickness of standing flange

i v=r Ic combined moment of inertia of stiffener and effective length of shell about axis

parallel to cylinder axis

111 moment of inertia of stiffener about its symmetric axis

J[L 2

L

n

ooc p

unsupported length of shell between stiffeners

length between rigid bulkhead supports

effective length of :shell between stiffeners

length between stiffeners

buckling mode circ:umferential wave number

axial load on shell ~a

out of circularity of the cylinder

external applied pressure

Bresse overall buckling pressure

collapse pressure of cylinder

design depth - corresponds directly to P

buckling pressure «lf endcap

collapse pressure o.f endcap

endcap yield pressure

pressures causing yield in shell, defined in section 6

pressure causing yield in ideal circular stiffener

vi

R

R

S.F.

T

w

v

w

pressure causing yield in stiffener including OOC effect

Bryant overall buckling pressure

minimized interframe buckling pressure

modified interframe buckling pressure

residual stress multiplication factor sinhg-sina ainha+sina

radius from cylinder axis to edge of standing flange

safety factor applied to load

At

thickness of faying flange of stiffener

-_ti ( u1 - u2)

( e sinh e cosh I - I cosh e sin f) sinh 2f:r: sin !/!(/sinh d cos I + e cosh e sin !) cosh 2f:r: cos !/! distance of stiffener centroid to inner edge of shell

3L"g-v2 ) a h 2

Poison's ratio

shell radial deflection

cosh a-cos a sinha+sina

A(l-f) A+bh+2Lh? .. yield stress of shell material

yield stress of stiffener material

frame stress at outer fiber

critical buckling stress of frame stiffener

Vii

1 Introduction

The structural analysis of a submarine pressure vessel consists primarily of the assessment of the pressures at which yielding and buckling occur. The shell, stiffener and bulkhead geometries and material properties can be varied to minimize the structure's weight and to avoid these failure mechanisms efficiently under a given design load. In theory, most submarine structural components are of simple geometry: cylinders, cones and hemispheres or torispheres. In reality, the submarine poses a very complex structural problem in that it deviates significantly from simplistic behaviour as a result of severe deviations from axisymmetric geometry (decks, cutouts, tanks, etc.) and fabrication effects (out of circularity, residual stresses, section connections, etc.). Today's engineers are fortunate to have access to powerful finite element programs to enable them to more accurately assess the probable behaviour of a particular design, and sophisticated test equipment to further verify analytical work. Unfortunately, these analyses come at a high price and can usually only be justified in final design stages.

Before the advent of these powerful tools, much work was concentrated on the development of manageable design formulae .. These formulae have been rigorously verified through experiment and comparison to more complex analyses methods1• Formulae methods have proven to be useful in preliminary design and comparison work, and form the basis of most code requirements such as the British Standard Specification for Unfired Pressure Vessels BS55002 • Where formulae have failed to explain real structural behaviour, extensive test data. have been employed to develop empirical methods of analysis.

In most cases, simplifying assumptions made to derive the design formulae limit their application. Fortunately, these limitations are generally within practical applications for submarine pressure hulls and are of a conservative nature. The use of safety factors accounts for analytical uncertainties as well as variation in material properties and load definition.

To facilitate the efficient application of these formulae, a computer program, PRHDEF, has been developed to introduce the structural parameters, change them easily and calculate the pressures associated with the various failure mechanisms. Safety factors are not included in any of the formulae with the exception of equation 22 for out of circularity described in section 2.8. They are instead calculated in relation to a given maximum design pressure. Therefore, input design pressures should not incorporate safety factors such as the 1.5 value used in BS5500. PRHDEF checks calculated safety factors against user specified values and Hags output values where desired safety factors are not met. PRHDEF determines failure loads for uniform axisymmetric hydrostatic pressure only. Variations in the load pattern resulting from underwater shock or collision will produce different failure mechanisms and pressure values and require more complex analyses using finite difference or finite element methods.

It is intended that this program be used for quick comparison of various design options

1

and as a basis for more complex analysis of submarine structure. PRHDEF should be particularly useful in preliminary comparisons of competitive design proposals. Due to variations of the design formulae and of the structural parameters used in the formulae in various codes, an indepen.dent analysis common to all designs is necessary for proper comparitive studies.

This report describes and evaluates the various formulae used in ring stiffened pressure vessel analysis and compare~~ results obtained with those of the BOSOR43 axisymmetric finite difference analysis program. Examples of the use of the program PRHDEF are given in Appendices A and B, whi<:h contain input and output data respectively.

2 Theory

2.1 Description of Submarine Geometry

The majority of the formulae incorporated in this study are for uniformly ring stiffened cylindrical shells of isotropi(: material. Formulae for assessing the collapse pressures of endcaps are also included. lrigure 1 shows the geometric information to be used for the cylinder, stiffeners and endca.p.

The length of the cylindElr is the distance between rigid ends. The rigid ends may be bulkheads or large frames which are much stiffer than the other frames in the section. Cylinders which terminate in. endcaps can be analyzed with an extended equivalent length of 0.4 times the endcap length2 • Sections with moderately varying stiffener spacing or dimensions can be analyzed using average values.

The stiffeners can be loeated on either the inner or outer surface of the cylindrical shell. For I section stiffeners the flange at the faying surface with the pressure hull may be included. For tee section stiffened shells the faying flange dimensions will be set to zero.

All material in the structure is assumed to have the same Young's modulus and Poisson's ratio; however, allowance has been made for different yield stresses in the shell and stiffener materials. The yield stress to be used is the design yield stress and the choice of this value is left to the user. There are no factors applied to the yield stress in the program as is done in the BS5500 code.

2.2 Determination of Effective Lengths of Shell Plating

In the determination of the moment of inertia of the combined stiffener and shell sections, and of the location, of the neutral axis, it is necessary to determine the length of shell within one bay which acts effectively with the stiffener in circumferential bending. The effect of shear lag in the shell plating and the stress distribution in the shell resulting from longitudinal bending between stiffeners results in a variation of the shell segment

2

contribution to stiffener bending. This variation can be compensated for by assuming a constant contribution over an effective length of shell plating.

Three options exist for determining the effective length in PRHDEF; all have been found in the literature in various submarine design formulae. The simplest method is to take 75 percent of the length between frames4 :

Le = 0.75L (1}

where L, the length between two frames is taken to be the frame spacing minus the web thickness or the faying flange width, whichever is less:

L = Lt- b or= Lt- hw (2)

The effective length is a function of the shell geometry in its deformed state and therefore is a function of the shell radius and thickness, the frame spacing and the circumferential wave number, n, of the buckling mode of interest. Bijlaard4 developed an expression for the effective length as a function of these parameters:

L _ 1.556VahfJ e- [·/1 + n~hll + nllh]i

V 2a11 V'3a (3)

An extensive set of tables (Figure 2} derived from reference 5 is used in PRHDEF which contains these tabulated data and an interpolation routine to determine effective length values as a function of the above mentioned parameters. These tables are used in the BS5500 code. A comparison of effective shell length values for various stiffened cylinder dimensions and circumferential wave numbers is given in Table 1.

2.3 Effects of Residual Stresses

Determining the total effects of residual stresses on the strength and stability behaviour of a pressure hull is an impossible task; however, it is known that the fabrication process for stiffened cylinders produces significant residual stresses in the shell and stiffeners of the order of 30 percent of the yield stress of the material. Residual stresses do not have a significant effect on the collapse loads of the shell1 but the determination of collapse through yielding in the stiffeners may be sensitive to residual stress effects. To account for this effect, the BS5500 code has adopted factors to be applied on the design load when determining stiffener failure. These values are 1.8 for hot formed or fabricated frames and 2.0 for cold bent frames. These factors do not address the major effect of residual stresses on failure through fatigue or environmentally assisted cracking. PRHDEF allows the user to input values of i for hot formed frames and 0.9(i) for cold formed frames. These are

3

the BS5500 values less the 1.5 general safety factor which allows determination of safety factors by PRHDEF. These values correspond to those used in reference 6.

2.4 Interframe Bifurcation Buckling Pressures Figure 3 illustrates the mode shape associated with interframe buckling. It generally

consists of half sine waves between stiffeners and several waves around the circumference. Von Mises7 developed a solutiion for the buckling of a simply supported cylindrical shell section which gives reasonable a.nd conservative predictions for interframe buckling of stiffened shells. There have been seveJ:al modifications of this work, two of which have been incorporated in PRHDEF. The VoiJl Mises expression as presented by Windenburg and Trilling8, minimized with respect to the circumferential wave number, n, is:

2.42E [ l!..j! P; - (l-v2)i 2a M- 1L . ( h ).1

2a - 0.45 2ii 2

(4)

The second version of thE! Von Mises expression and the one used in the BS5500 code is that presented by Kendricks1~:

Eh { 1 h2

[ 2 ("'"a)2]} PMl = ..,..a-[n_2 ___ 1_+_2!:1:j;;] [n2(:a)2 + 1]2 + 12a2(1- v2) n - 1 + L (5)

To find the minimum pre11sure with respect to n, an iterative solution is required varying n in the range of 2 to 20 wHh most geometries producing minimums in the 10 :::; n :::; 16 wave range. Equation 5 is m«Jdified to be more accurate for low wave numbers by including the n2-1 term. A comparison of equations 4 and 5 is given in Table 2.

2.5 Determination of Stress Values

The hydrostatically load13d stiffened cylindrical section of a submarine is subject to both lateral and axial pressure. The differential equation for the lateral deflection of a cylinder subject to hydrostatic pressure is given as:

84w pa o2w Eh II D a~• + 2 a~2 + a2 w = p(l- 2) (6)

The second term arises frc1m the inclusion of axial load and results in a nonlinear pressure vs deflection relationship. The degree of nonlinearity increases rapidly as the axisymmetric buckling pressure of the cylinder is approached. At pressures well below the axisymmetric buckling pressure the degree of nonlinearity is small. Since submarines are designed to reach their yield strength well before reaching the axisymmetric buckling pressure (see section 2.8)

4

the pressure vs defiection relationship is nearly linear for the area of interest to submarine designers. Wilson1 took advantage of this fact and derived expressions for stresses in the critical regions of the structure neglecting the second term of equation 6.

The regions where stresses are of concern are:

• n3 - the maximum circumferential stress in the outer fiber of the shell at midbay,

• 0'&- the mean circumferential stress in the shell at midbay,

• n1 - the longitudinal stress in the shell on the inside surface at the stiffener connection,

• O'Jfl - the circumferential stress in the standing fiange of the stiffener.

The expressions presented by Wilson for these stresses, rearranged to give external pressures which cause yielding to occur, are:

Ps = hn" ( 1 ) . a 1+1H

(7)

R = hn11 ( 1 ) & a 1 +,a (8)

p1

= 2hn11 ( 1 a [1 + ( 1~!::i)!]IR

_ hn111A1 A PFY- a2(1- i)[1 + bh+ ¥] (10)

where A,,, H, G, R, fJ and a are defined in the notation. A more realistic determination of the pressure causing yield in the midplane of the shell

at midbay is found using the Hencky-Von Mises yield criteria. This is determined by:

(11)

This results in higher allowable pressures than consideration of the circumferential stress value Pc&·

Salerno and Pulos10 developed solutions for the stress including the effect of axial pressure (ie. not neglecting the second term of equation 6). These solutions are nonlinear functions of the pressure, P, and therefore an iterative solution is required. Again the PRHDEF code solves for the pressures to cause yielding at the critical locations in the shell:

5

h hH 2(1 - v2)L2 1 Pu =a-,[·-;+ a(1- j)TU0 - ( va2(1- j) )TJoL2 ]

(12)

h hH PSA = u,[-; + a(1- j)TU] (13)

2h 2(1 - v2)L2 1 Pu = u,[-; + ( a2(1- j) )TJfL2] (14)

Eh GU u11 {d PFYA = a2 ( 1 - j) ( H - ---:E) (15)

where T ,Jo, J 1.: ,U ,G and H a.re defined in the notation. It is expecled that for most realistic submarine geometries, the two methods of deter

mining stress will give comparable results. Table 3 gives a comparison of the two methods for models A, B and C.

2.6 BS5500 Collapse Curve for Cylinders

The determination of 1;he elastic interframe buckling pressure (section 2.4) and of the pressure at which yield is reached (section 2.5) is for ideal stiffened cylinders. In experimental studies1 , interaction between plastic collapse and elastic buckling has given significant scatter to collapse pressure data. Upper and lower bound curves for all available experimental data established a relationship between Pfo!' and k used in BS5500 and reproduced in Figure 4. PM1 is defined in equation 5, Ps is defined in equation 8 and Pc is the collapse pressure of the cylinder. A simplified expression for the lower bound curve of Figure 4 has been coded int1> PRHDEF":

P Ps -:::1--Ps 2PM1

p;,:l < 1.0

This is accurate to within 1% of the lower bound curve of Figure 4.

(16)

The BS5500 code uses this curve by applying a safety factor of 1.5 to the lower bound curve. The 1.5 safety factor has not been included in the PRHDEF calculation so that a safety factor can be calculated for the analyst to use in his own criteria.

The curve of Figure 4 is limited to cylinders with a maximum out of circularity of 5% on radius and for cylinders of 5.9 < I < 250 and 0.04 < ~ < 501. These limits result from

6

the range of experimental data surveyed in forming the curve and are applicable to most submarine dimensions. PRHDEF checks input data for these dimension limits and flags violations.

2. 'T Overall Buckling Pressure

Figure 5 shows the mode shape for overall buckling of a ring stiffened cylinder. It consists of a half sine wave between rigid ends and usually 2 to 6 waves circumferentially. The circumferential wave number depends on the length to radius ratio of the cylinder.

Two expressions for overall buckling have been incorporated in PRHDEF. Both of these are dependent on the wave number, n, and are presented for wave numbers 2 to 6.

Bresse7 developed an expression for an infinitely long stiffened shell:

P ( ) _ (n2 - 1)El0

B n- 3 aL (17)

where [0 is the moment inertia of a combined shell and stiffener section and is therefore sensitive to the effective length of shell chosen (section 2.3). This formula greatly underestimates overall buckling loads for finite lengths of shell supported by rigid bulkheads. The membrane shear stresses that occur in a realistic stiffened shell are not accounted for by equation 17. This inaccuracy decreases with increasing length of shell and circumferential wave number.

Bryant11 developed an approximate equation for the overall buckling load by combining equation 5 for the buckling of the shell between rigid ends and equation 17 to incorporate the effect of the stiffeners:

P. (n) _ Eh ~~. 1 (n2 - 1)Elc

n - a [n2 -1 + !(1~)2][(~)2 + 1]2 + aSL (18)

It has been shown that this formula gives good results in comparison with more complex theories; however, it gives unconservative results for cylinders which fail with circumferential wave numbers greater than 3, as would be the case for short stubby sections. Table 4 compares formulae 17 and 18 with numerical results from BOSOR4.

A formula for the axisymmetric collapse load of the stiffened cylinder is included in PRHDEF. This mode of failure usually occurs at much higher loads than non-axisymmetric modes (equations 17 and 18), but is useful in determining the degree of nonlinearity present in the yield stress calculations (section 2.6). The axisymmetric buckling load equation is6 :

(19)

7

2.8 Out of Circularity and Frame Collapse

Equations 17 and 18 give overall buckling loads for ideal stiffened cylinders. In practice, these values are difficult to attain as the structures or experimental models are not ideal. Out of circularity (OOC) of the cylinder causes reduction of the ideal buckling loads particularly if the OOC is in the form of the critical buckling mode shape. The bending stress induced in the stiffener as a result of OOC greatly reduces the pressure at which it reaches yield. The effects of OOC are incorporated in design by determining the failure of an out of round stiffener andl assumming that stiffener failure precipitates overall buckling collapse. The stress in a stiffener of a shape corresponding to the critical overall buckling mode as a result of compress:ion and bending is9 :

(20)

where Cn is the out of ciJ~cularity. To determine the pressure Pf, at which the stiffener will reach yield including OOC

effects, equation 20 must be rewritten in the form:

This quadratic is then solved for P1 as a function of n. The BS5500 code uses a11 allowable OOC of 5% of the radius. Cn can be replaced by

.005a in equations 20 and 21 and the residual stress factor, R set equal to 2 (which includes the 1.5 safety factor in addition to the i residual stress factor) to get the form found in BS5500.

Since u 6 J is a function of the wave number, n, an alternative approach is to determine the OOC which will cause yiE1ld (u6J = u111) in the stiffener for a given n. The determination of Cn will not result in the c:alculation of a safety factor since it is a nonlinear function of the applied load, P. Therefo:re, in this case the applied load has been multiplied by 1.5 to give Cn including a safety factor. Rewriting equation 20 to solve for Cn as a function of n, gives:

(22)

8

----------------------------------------------------------------------------------·

PRHDEF determines On for wave numbers 2 to 6 using equation 22. Frame failure loads are calculated using equation 21 with the value of On being derived from equation 22, given a value of 0.005a. from BS5500, or given a. user specified OOC value. To determine On for a different safety factor than 1.5, R ca.n be input such that R = ~{ (~). It should also be noted that the frame failure load Pt predicted by PRHDEF using On from equation 22 will be equal to the input design pressure times the 1.5 sa.fety factor.

2.9 Stiffener Buckling

Two types of buckling failure have been discussed in previous sections: interfra.me a.nd overall buckling. A third possible mode of buckling failure can occur in the ring stiffeners from lateral torsional buckling (tripping) or local buckling of the web or flange. Interaction of the stiffener with the shell plating requires more sophisticated analyses such as finite element or finite difference for the determination of the loads ca. using this mode of failure. Since stiffener buckling is easy to avoid by proper dimensioning of the stiffeners, simple conservative formulae have been introduced in BS5500 to check stiffener dimensions.

A formula. for determining the torsional buckling of a T stiffener which is pinned a.t its connection to the shell is given asl:

Elz tier = < t111f

A1r1z1

Dimensions for the web and flange a.re checked respectively by2 :

2.10 Endcap Collapse

dw 'f l.l [iff" hw V ;;;J

(23)

(24)

(25)

Integrity of endca.p structure is checked in a. similar manner to cylinders using a.n exper" imentally derived collapse curve (Figure 4). Hemispheres, spheres and torispheres (using the outer radius) are analyzed by the same method. The pressure causing yield in the shell is determined from:

he Pe11 = 2-t111 ae

The elastic critical buckling pressure for a. perfect sphere is determined from1:

9

(26)

P. _ 2Eh~ ec - a~J3(1 - v2) (27)

The ratio ~ is then used to determine k'- from Figure 4. PRHDEF uses a tabulated form of Figure 4 to determinte Peu· This experfmental curve is limited to endcaps which do not vary in radius by more than 1 percent.

3 Discussion

Classic formulae used to check the integrity of a stiffened pressure vessel design against various failure mechanisms lhave been described. The methods of the BS5500 code for externally loaded pressure vessels include most of these formulae. Design curves incorporating experimental data 1to derive the collapse load have also been given. The results obtained by these methods are subject to the geometry of the structure. The computer code PRHDEF has been written ·to incorporate the formulae described in the previous section and enable easy input of structural 4ata. Appendix A contains a sample terminal session with PRHDEF and Appendix B contains samples of PRHDEF analysis for three models. Original geometry data are 21tored on a file which can be used by PRHDEF in subsequent runs to study the effects of v1uying one or several structural parameters. PRHDEF is a self explanatory interactive Fortran computer code operational on DEC-20, VAX 11/750 and IBM PC computers.

Three options of determining the effective length of shell segment to be included in determining equivalent moments of inertia of the stiffener for circumferential bending have been given. Table 1 compar,es these options for three models. As would be expected, the larger the stiffener spacing and the thinner the shell, the smaller the effective length. The effective length of 0. 75L 1 wan not very satisfactory for models B or C when compared to the other methods. The second t.wo options which include the effect of the circumferential wave number n, show decreasing .~11 with increasing n. Both of these methods give comparable results, with the tables used in the BS5500 code (derived from reference 4) being most conservative.

Table 2 gives results of iuterframe buckling pressures for the three models described in Table 1. The minimum values from equation 5 agree well with those obtained from equation 4. All of the minimum values from the formulae are less than the numerical results for the three models. This is expect•3d as the formulae consider only one bay of shell with a simply supported boundary condition at each end. The full structure of several bays provides some rotational restraint at 1;he frame supports which results in a stiffer structure and thus higher buckling loads. This is particularly evident in model A which has a thicker shell. For practical purposes, the minimized Von Mises expression of equation 4 appears to give

10

adequate results. Table 3 compares results obtained for the three models for the linear and nonlinear stress

values. The value of the axisymmetric buckling pressure is also included in the table as it indicates the degree of nonlinearity expected in the pressure - stress relationship. Model B shows yield pressures which approach half of the axisymmetric buckling load and some difference between the linear and nonlinear values is present. The differences agree with the trend indicated in Figure 61; ie. nonlinear decrease in shell stress and nonlinear increase in stiffener stress as the pressure approaches the axisymmetric buckling pressure. These stress values agree well with the BOSOR4 results.

Table 4 compares the overall buckling loads obtained by equations 17 and 18 and numerical results with BOSOR4 for the three models. The results of equation 17 for a section of shell and frame alone (infinite cylinder) are too low for small wave numbers. This is a result of the increased stiffness of the cylinder from localized membrane shear stresses at the boundaries. This effect decreases with increasing length to radius ratios. Equation 18 neglects membrane shear stiffness from interframe deformation which results in buckling loads which approximate those of a stiffer cylinder. This effect decreases as the length increases and as the minimum wave number approaches 2. The numerical results should converge to a better approximation of the true buckling mode for stubby ring stiffened cylinders. The BOSOR4 models had simply supported boundary conditions and were fixed axially at one end. Variation of these boundary conditions drastically alter the buckling loads and mode shapes.

The BS5500 code method of determining frame failure is very strict. All models had this as their lowest failure load. This approach is generally viewed to be pessimistic; however, it has proven to be the most practical approach to incorporating the OOC effects on the overall buckling load collapse mechanism.

The stiffener tripping criteria are also recognized as being pessimistic. Model C which takes stiffener dimensions from an actual submarine fails this criteria. In this event more complex analysis with a program such as BOSOR4 which will take into account the interaction between the stiffener and the shell may prove that the stiffener is more than adequate.

4 Conclusions

The use of formulae and the BS5500 pressure vessel design code to evaluate hydrostatically loaded, uniformly stiffened cylinders has been presented. This approach relies on the structure resisting three mechanisms of failure. These are: interframe collapse which depends on the interframe bifurcation buckling pressure and the shell yield strength; overall collapse which depends on the stiffener yield strength and the overall bifurcation buckling pressure including the effects of out of circularity; and localized stiffener failure. The

11

loads associated with these failure mechanisms can be quickly determined for a variety of geometries with the computE~r code PRHDEF which accompanies this work.

The simplified formulae compare well with BOSOR4 numerical calculations for the three models considered with the exception of the overall buckling formula (equation 18) which significantly overestimates the buckling load for short cylindrical sections. Better approximations to the stress, interframe and overall bifurcation buckling loads and local stiffener failure may be obtained by using finite element or finite difference methods, the results of which may be applied to the design curve (section 2. 7) and equation 21 to determine overall collapse. Geometries which differ significantly from uniformly stiffened cylinders should be analyzed with numerical methods. For preliminary design or comparitive studies, the formulae presented here and incorporated in PRHDEF should be adequate for most cylindrical submarine pressure vessel sections.

12

TABLE 1: Comparison of Effective Lengths(mm) for Various Formulae

Wave Model A4 Model Bb Model cc Number 0.75LJ Eqn 3 BS5500 0.75LJ Eqn 3 BS5500 0.75LJ Eqn 3

2 375 421 413 600 472 459 570 442 3 375 409 393 600 468 453 570 437 4 375 392 371 600 464 444 570 430 5 375 371 346 600 457 430 570 422 6 375 347 319 600 450 415 570 411

4 Model A: h = 40mm .LJ = 500mm dw = 150mm hw = 15mm WJ = 150mm h1 = 15mm Lb = lOOOOmm a= 2000mm u11 = 500MPa u111 = 450MPa v = 0.3 E = 207000M Pa R = 1.2 Pd = 500m internal framing

11 Model B: h = 20mm L1 = 800mm dw = 150mm hw = 15mm Wf = 150mm hJ = 15mm L, = 8000mm a= 4000mm ufl = 600~Pa Uuf = 450MPa v = 0.3 E = 207000MPa R = 1.2 Pd = 125m internal framing

c Model C: h = 23mm LJ = 760mm dw = 203mm hw = 6.1mm WJ = lOlmm hJ = 8.4mm Lb = 13680mm a= 3086mm u11 = 450MPa u111 = 450MPa v = 0.3 E = 207000M Pa R = 1.0 Pd = 200m internal framing

13

BS5500 429 419 406 389 371

·TABLE 2: Comparison of Interframe Buckling Pressures (MPa)

Wave Model A Model B Model C Number Plf PMt :BOSOR4c PM PMt BOSOR4 PM PMl BOSOR4

6 .• 76.1 74.2 6.1 3.8 11.5 7.2 7 72.4 74.3 5.4 3.6 9.8 6.7 8 69.8 75.8 4.7 3.3 8.3 6.2 9 68.3 78.5 4.2 3.1 7.2 5.8 10 70.5 67.9 83.2 3.7 3.0 6.3 5.5 11 68.5 88.9 3.3 2.9 5.7 5.2 12 69.9 93.7 3.0 2.8 5.3 5.1 13 72.0 98.8 2.7 2.7 5.0 5.0 14 74.7 104.2 2.5 2.6 4.8 5.1 15 78.1 109.8 2.4 2.5 4.8 4.8 5.1 16 81.9 115.5 2.3 2.5 4.8 5.3 17 86.2 121.4 2.23 2.50 4.9 5.4 18 90.9 127.4 2.21 2.51 5.0 5.6 19 96.1 133.6 2.22 2.20 2.53 5.2 5.9 20 101.6 140.0 2.23 2.6 5.5 6.1

a equation 4, b equation 5, c reference 3

TABLE 3: Comparison of Linear and Nonlinear Yield Pressures (MPa)

Model PG i1A 11 PfA PJ P/A PJ.y PPvA P! A 11.4 11.3 11.9! 11.8 13.4 13.2 12.5 12.5 221.1 B 3.0 2.7 3.3 3.0 2.9 2.5 3.8 5.2 7.8 c 3.4 3.3 3.5 3.4 4.2 4.0 4.5 4.6 16.1

a Wilson equation 7, 6 Salerno :md Pulos equation 12, c Wilson equation 8 d Salerno and Pulos equation 1:3, e Wilson equation 9, I Salerno and Pulos equation 14 11 Wilson equation 10, h SalerJuo and Pulos equation 15, i Axisymmetric buckling load equation 19

14

TABLE 4: Comparison of Overall Buckling Pressures (MPa)

Wave Model A Model B Model C Number P:: .!'B BOSOR4c Pn PB BOSOR4 Pn PB BOSOR4

2 22.8 12.4 25.0 36.3 0.7 3.9 7.4 1.5 5.2 3 33.6 32.7 36.7 7.1 1.9 3.1 4.4 3.9 4.9 4 60.7 60.5 54.2 4.8 3.6 3.5 7.3 7.2 6.8 5 95.4 95.4 67.9 6.1 5.8 3.9 11.5 11.4 7.4 6 136.6 136.6 73.2 8.4 8.3 3.9 16.6 16.6 7.2

11 Bryant equation 18, b Bresse equation 17, c reference 3

15

r---;: Lf~ I r.-*·1 hl I

Nx__....l h~b"*~ ~f~

r1 ° d I _..__I _........__I -lr.--~ _ ____.""""'---

Figure 1: Geometry of Cylinder and Endcap Section

16

~ ~w-• 12R~ "'

L. I I L

'

~ 2 R

0 1.0980 0.01 1.08:23 0.02 1.0663 0.03 1.050-l 0.04 0.9907 0.05 0.8976 0.06 0.7921 0.07 0.6866 0.08

.. 0.6111

0.09 0.5355 0.1 0.4600

12R' = 10-•

Lc/ . L,

L, n 2 :?.:tR

0 1.0980 0.01 ,, .0823 0.02 1.0345 0.03 I o.9019 0.04 0.7242 0.05 j 0.5602 0.06 0.4483 0.07 0.3752 0.08 0.3263 0.09 0.2920 0.1 0.2578

s

3 14 5 6 ~12 2rrR

-1.0980 1.0980 1.09SO 1.0980 1.0~23 1.0663 1.0663 1.050-l 1.050-l 1.0~65 0.9947 0.9629

.J.oo:n I 0.9549 0.9019 0.8435 0.9231 0.8515 0.7838 0.708:! 0.8276 0.7512 0.6716 .0.5952 0.7:!93 0.6609 0.5871 0,5143 0.6321 0.5707 0.5025 0.4343 0.5630 0.5088 0.4480 0.3877 0.-1940 0.4470 0.3935 0.3410 0.4249 0.3852 0.3390 0.2944

0 1.0980 0.01 0.9072 0.02 0.4297 0.03 0.1759 0.04 0.::!207 0.05 0.165.5 0.06 0.1490 O.D7 0.1324 0.08 0.1159 0.09 0.0993 0.1 0.0828 ~

e: 10- • !2R• =

•

~-~ 15 I 6 I

i 1.09SO ·--

I 1.0980 ! .1.0980 ! 1.0980 !.OS2J I !.0663 1 1.o66J ,1.0504 !.OJ S6 0.9947 I o.96::!9 I 0.9311 0.8807 0.8541 l 0.8117 0.7639 0.7003 I o.6:'24 I 0.6326 0.5929 0.5411 1 o.s:!2o 0.49.34 I 0..164:-0.4350 i 0.4218 I 0.4005

I 0.Ji9J I 0 ''i'\6 0.3661 I o.3547 0.3388 I ·"-"

0.3163 I 0.3084 0.296-l I o.::sos 0.:2847 0.2775 I o.::66o 0.~:~

0.2531 0.:2467 0..2355 0 ...,_.,. ... I

___ .... ..,

~ 2 2rtR . . - .

0 1.0980 0.01 1.0663 0.02 0.8276 0.03 0.5252 0.04 0.3740 0.05 0.2960 0.06 0 . ..,661 0.07 0.2362 0.08 0.:063 0.09 0.1763 0.1 0.1464

Figure 2: Effective Length Tables2

17

3 4 5 6

1.0980 1.0980 1.0980 1.0980 0.9072 0.8913 0.8913 0.8913 0.4297 0.4218 0.4218 0.4218 0.::!759 0.2759 0.2759 0.2759 0.2207 0.2207 0.2191 0.2191 0.1655 0.1655 0.1623 0.1623 0.1487 0.1487 0.1461 ; 0.1461 0.1318 0.1318 0.1299 0.1299 0.1149 0.1149 0.1136 0.1136 o:o98o 0.0980 0.0974 0.0974 0.0812 0.0812 0.0812 0.0812

3 14 /s 6

1.0980 1.0900 1.0980 l.09SO 1.0504 1.0504 J.G504 1.0345 0.8196 0.8037 0.7878 0.7719 0 . .5199 0.5146 '0.5040 0.4934 0.3700 0.3661 0.3621 0.3541

'0.2928 0.2897 0.2865 0.2801 0.2632 0.2604 0.2575 0.2521 0.2336 0.231 I 0.2285 0.2241 0.2040 0.2018 0.1996 0.1961 0.1744 0.1725 I 0.1706 0.1681 0.1448 0.1432 0.1416 0.1401

LONGITUDINAL CIRCUMFERENTIAL

Figure 3: Interframe Buckling Mode2

1.2 r-·---.---,..---r--,..--........ ---,-----,

1.0

Pc .8

p5 or .6

Peu

Pey .4

0

'/

,,/ ,..'\..,UPPER BOUND

,..""' SPHERES ------/ ......... / ....... , ....... , .,..,. , ,.,. , "

.,/ /""'-LOWER BOUND 7 / SPHERES /

•'

2 3 4 5 PMt Pee -or-p5 Pey

6

Figure 4: Collapse Pressure Design Curve8

18

7

LONGITUDINAL CIRCUMFERENTIAL

Figure 5: Overall Buckling Mode Shape

w 0:: ::::> C/) (/) w 0:: a..

p

RADIAL DEFLECTION

AXISYMMETRIC BUCKLING PRESSURE

Figure 6: Stress vs Pressure Relationship1

19

Appendix A: Sample Terminal Input Session of PRHDEF

COMPILE SOURCE IF NECESSARY $ FOR PRHDEF

LINK .OBJ FILE TO GET .EXE FILE $ LINK PRHDEF

$RUN PRHDEF

PROGRAM PREDICTS THE STRENGI'H OF FRAME STIFFENED CYLINDERS AND ENDCAPS

TO RUN EXISTING DATA FILE ENTER 1 TO GENERATE A NEW MODEL ENTER 0 0

ENTER A FIVE CHARACTER PREFIX NAME TO IDENTIFY NEW OR OLD DATA FILE MOD LA

FOR OU'IPUT TO SCREEN ENTER 5 FOR OU'IPUT TO FILE PREFX.OUT ENTER 3 3

INPUT TITLE BLOCK - 70 CHARACTERS MAXIMUM

MODLA TERMINAL SESSION ENTER ALL DATA IN CONSISTENT UNITS

IN.=.25.4MM. 1 PSI.= 0.00689 MPa 1 METER=39.3701IN. MOMENT OF INERT! IN**4 = 41.623CM**4 IN**4 = 416.231MM**4

YOUNGS MODULUS STEEL = 30000000 PSI .207000 MPa ALUM = 10300000 PSI 70000 MPa

YIELD STRESS STEEL = 45000 PSI :no MPa ALUM = 39000 PSI .270 MPa

INS. OR MM ARE THE MOST FAVOURED UNITS CHOOSE FROM FOLLOWING~ UNITS FOR THE ANALYSIS INCHES = 1 MILIMETERS = 2 FEET = 3 METERS = 4 2

ENTER THICKNESS OF CYLINDER SHELL PLATING 40 ENTER FRAME SPACING 500 ENTER WIDTH OF FAYING FLANGE IN CONTACT WITH PLATING 0 ENTER THICKNESS OF FAYING FLANGE 0 . 0 FOR WELDED TEE BAR 0 ENTER DEPTH OF FRAME WEB 150

20

ENTER THICKNESS OE E'RAME WEB 15 ENTER WIDTH OE E'RAME ELANGE 150 ENTER THICKNESS OE E'RAME ELANGE 15

* * INPUT DATA * * 1 THICKNESS OE PLATING 40.0000 MM.

2 FRAME SPACING 500.0000 MM.

3 WIDTH OE EAYING ELANGE IN CONTACT WITH PLATING 0. 0000 MM.

4 THICKNESS OE EAYING ELANGE 0. 0000 MM.

5 DEPTH OE E'RAME WEB 150.0000 MM.

6 THICKNESS OE FRAME WEB 15.0000 MM.

7 WIDTH OE E'RAME INNER ELANGE 150.0000 MM.

8 THICKNESS OE FRAME ELANGE 15.0000 MM.

TO CORRECT A DATA ERROR ENTER LINE NO TO CONTINUE ENTER 0 0

ENTER RADIUS OE MEAN SUREACE OE CYLINDRICAL SHELL PLATING 2000 ENTER DISTANCE BETWEEN RIGID ENDS 8000 ENTER YIELD STRESS OE SHELL PLATING 500 ENTER YIELD STRESS OE FRAME ELANGE 450 ENTER POISSONS RATIO .3 ENTER YOUNGS MODULUS 207000 ENTER 1. 0 EOR INTERNAL FRAME OR -1. 0 EOR EXTERNAL E'RAMING 1 ENTER OUT OE CIRCULARITY - A MINUS VALUE WILL USE BS5500 CODE -1 CHOOSE MULTIPLIER EOR RESIDUAL STRESS IN FRAMES NO ALLOWANCE EOR RESIDUAL STRESS = 1 MULTIPLIER EOR COLD BENT FRAMES= 1. 33 MULTIPLIER EOR HOT EORMED E'RAMES = 1.2 1.2

1 RADIUS OE MEAN SUREACE OE SHELL PLATING 2000.00 MM.

2 DISTANCE BETWEEN RIGID ENDS 8000.0 MM.

3 YIELD STRESS OE SHELL PLATING 500. MPa

4 YIELD STRESS OE E'RAME ELANGE 450 . MPa

5 POISSONS RATIO 0.300

21

6 YOUNGS MODULUS 207000. MPa

7 FRAMING TYPE 1. INTERNAL FRAMING

8 OUT OF CIRCULARITY -1.000 MM.

9 MULTIPLIER FOR RESIDUAL FRAME STRESS 1.20

RADIUS THICKNESS RATIO a/h 50.00

LENGTH RADIUS RATIO L/a 4.000

TO CORRECT A DATA ERROR ENTER LINE NO TO CONTINUE ENTER 0 2 ENTER DISTANCE BETWEEN RIGID ENDS 10000

1 RADIUS OF MEAN SURFACE OF SHELL PLATING 2000.00 MM.

2 DISTANCE BETWEEN RIGID ENDS 10000.0 MM.

3 YIELD STRESS OF SHELL PLATING 500. MPa

4 YIELD STRESS OF FRAME FLANGE 450. MPa

5 POISSONS RATIO

6 YOUNGS MODULUS

0.300

207000. MPa


8 OUT OF CIRCULARITY -1.000 MM.

9 MULTIPLIER FOR RESIDUAL FRAME STRESS 1. 20

RADIUS THICKNESS RATIO ajh 50. 00



ENTER ENDCAP THICKNESS - ENTER 0 FOR NO ENDCAP 40 ENTER ENDCAP RADIUS ·- OUTER END RADIUS FOR TORISPHERE 2000

1 THICKNESS OF ENDCAP 40.0000 MM.

2 RADIUS OF EN[X;.AP 2000.0000 MM.

TO CORRECT A DATA ERROR ENTER LINE NO TO CONTINUE ENTER 0 0 ARE METRIC UNITS USED YES = 1 NO = 0

22

1 ENTER MAXIMUM -SURVIVABLE- DESIGN DEPTH IN METERS 500 ENTER SAFE'IY FACTORS FOR YIELDING - INTERFRAME BUCKLING

AND OVERALL BUCKLING- VALUES OF 1.5 2.5 AND 3.5 ARE TYPICAL THESE VALUES ARE USED FOR COMPARITIVE PURPOSES ONLY AND NOT IN CALCULATION

1.0 2.5 3.5

1 SPECIFIED MAXIMUM DEPTH 500.0 M.

MAXIMUM DESIGN PRESSURE 5.000 MPa

2 SAFE'IY FACTOR FOR STRESS COMPARISONS - 1. 50 SAFE'IY FACTOR FOR INTERFRAME BUCKLING COMPARISONS 2. 50 SAFE'IY FACTOR FOR OVERALL BUCKLING COMPARISONS - 3.50


ENTER FORM OF EFFECTIVE LENGTH OF SHELL BE1WEEN STIFFENERS FOR USE IN CALCULATIONS --1=0.75XFRAME SPACING 2=BIJLAARDS FORMULA - SEE REPORT ON PRHDEF 3=BS5500 TABLES FOR EFFECTIVE LENGTH 3 CALCULATION FINISHED

23

Appendix B: Sample Output Analysis of PRHDEF for Three Models

MODEL A

.. INPUT DATI' ••

THICKNESS OF PLAT!~; 40.0000 MM.

2 FRAME SPACING 50e.E1000 W.

J WIDTH OF rAYING F~~E IN CONTACT WITH PLATING 0.0000 MM.

4 THICKNESS OF FAYING FLANGE 0.0000 MM.

5 DEPTH OF FRAME WEB 150.0000 MM.

6 THICKNESS OF FRAME WEB 15.0000 t.IA.

7 WIDTH OF FRAME INNER FLANGE 150.0000 t.IA.

8 THICKNESS OF FRAMC FlANGE 15.0000 t.IA.


2 DISTANCE BETWEEN RIGID ENDS 10000.0 t.IA.

3 YIELD STRESS OF SHELL. PLATING 509. 1.4Pa

4 YIELD STRESS OF FRAME: FLANGE 450, MPa

5 POISSONS RATIO

6 YOUNGS t.«:JJULUS

0 . .300

207000. MPa


8 OUT OF CIRCULARITY -1.000 ......

9 MULTIPLIER FOR RESIDUAL FRAME STRESS 1.20

RADIUS THICKNESS RATIO a/h 50.00


THICKNESS OF ENDCAP 40.0000 t.IA.

2 RADIUS OF ENDCAP 2000.0000 t.IA.



2 SAFETY FACTOR FOR STRESS COMPARISONS - 1.50 SAFETY FACTOR FOR !NTERFI~AME BUCKLING COMPARISONS - 2.50 SAFETY FACTOR FOR OVERALL BUCKLING COMPARISONS - 3.50

24

-

EFFECTIVE LENGTH OF SHELL IS FROM 855500 TABLES

••••••CALCULATED OUTPUT••••••

INTERFRAME COLLAPSE

VON toUSES ELASTIC INTERFRAME BUCKLING PRESSURE Pt.4 WINOENBURG AND TRILLING

KENDRICKS MINIMIZED MODIFIED VON MISES ELASTIC INTERFRAME BUCKLING PRESSURE AT WAVE NI.M3ER- 10

PRESSURE

70.512 I.Fa

67.916 I.Fa

BS5500 LONER BOUND COLLAPSE CURVE GIVES P .. f1/PY-5. 73 PC/PY-0.91 AND COLLAPSE PRESSURE • 10.812 I.Fa

••• SHELL YIELDING

HOOP STRESS FOR AN UNSTIFFENED CYLINDER PB

PRESSURE WHERE CIRCIA.IFERENTIAL STRESS EQUALS YIELD STRESS IN PLATING P3 -WILSON LINEAR FORWLATION

PRESSURE WHERE MAX. LONGITUDINAL STRESS IN THE PLATING EQUALS THE YIELD STRESS P7 - WILSON

PRESSURE WHERE MEAN MIDBAY CIRCIA.IFERENTIAL PLATING STRESS EQUALS YIELD STRESS PS - WILSON

PRESSURE WHERE MEAN MIDBAY STRESS IN PLATING WITH HENCKY-VON MISES YIELD CRITERION EQUALS YIELD

n• FRAME YIELDING

PRESSURE WHERE CIRCUMFERENTIAL STRESS IN THE STANDING FLANGE EQUALS YIELD PFY - WILSON

••• ~E ACCURATE ITERATED SOLUTION FOR STRESS SALERNO AND PULOS NONLINEAR FORMULATION - SHELL

PRESSURE AT WHICH THE MAXIMUM CIRCut.IFERENTIAL STRESS IN THE PLATING EQUALS THE YIELD STRESS P3A

PRESSURE AT WHICH THE LONGITUDINAL STRESS IN THE IN THE PLATING EQUALS THE YIELD STRESS P7A

PRESSURE AT WHICH THE CIRCUMFERENTIAL STRESS IN PLATING AT MIDBAY EQUALS THE YIELD STRESS PSA

•••FRAME STRESS

PRESSURE WHERE CIRCUMFERENTIAL STRESS IN STANDING FLANGE OF THE FRAME EQUALS YIELD - PFYA

25

10.000 ~.Fa

11.353 I.Fa

13.376 I.Fa

11.845 ~.Fa

13.601 ~.Fa

12.494 MPa

11.317 MPo

13.199 MPa

11 .826 MPa

12.513 MPo

SAFETY FACTOR

14.1025

13.5833

2.1625

2.0000

2.2706

2.6751

2.3691

2.7202

2.4988

2.2634

2.6397

2.3653

2.5025

•••BS5500 ENDCAP RESULTS

PRESSURE AT WHICH ENDCAP REACHES YIELD 20.000 t.Fa

CRITICAL BUCKLING PRESSURE OF' ENDCAP 100.188 MPa

ULTIMATE COLLAPSE PRESSURE FROM BS5500 P/PY CURVE THIS IS THE LOWER BOUNCI CURVE OF' EXPERIMENTAL DATA 9.989 t.Fa

•••OVERALL COLLAPSE

SYMMETRIC BUCKLING PRESSURE OF' THE RING-SHELL CCM31NATION - WAVE NlM3ER 0 t.IODE

BRYANT OVERALL BUCKLING WITH EFFECTIVE WIDTH BRESSE SOLUTION IS FOR A SINGLE STIFFENER AND SHELL SECTION ASSUMMING SHELL FULLY EFFECTIVE

221.147 t.Fa

COLLAPSE PRESSURE t.«:>DE EFFECTIVE LENGTH OF SHELL BRESSE BRYANT

2 412.894 12.38-4 22.828 t.Fa

3 392.893 32.688 33.579 t.Fa

4 370.895 60.542 60.700 t.Fa

5 345.871 95.383 95.42-4 t.Fa

6 319.448 136.573 136.587 t.Fa

~·•CHECK ON STIFFENER PRC~TIONS FROM BS5500

CRITICAL TRIPPING STRESS: OF STIFFENER -THIS IS 1.8923 TIMES YIELD

851.5-40 t.Fa

GENERAL STIFFENER PROPORTIONS 0. 00411 WITHIN CODE RECOMMENDATIONS 0.00217

WEB DEPTH TO THICKNESS R~TIO 10.000 IS WITHIN CODE REcot.t.IENOATIONS 23.592

HALF FLANGE WIDTH TO THICKNESS ~TIO 5.000 IS WITHIN CODE RECOMMENDATIONS 10.724

26

-4.0000

20.0376

1.9977

44.229-4

-4.5655

6.7158

12.1399

19.0848

27.3173

•••STIFFENER FLANGE FAILURE

TWO VALUES OF OUT OF CIRCULARITY ARE USED TO DETERMINE THE FAILURE PRESSURE OF THE STIFFENER - KENDRICKS FORMULA WHICH IS A FUNCTION OF WAVE NUMBER AND OVERALL BUCKLING LOAD AND EITHER THE BS5500 VALUE OR A GIVEN OOC VALUE, BOTH OF WHICH ARE CONSTANT WITH WAVE NUt.13ER.

EFFECTIVE OUT OF ROUND FAILURE OUT OF ROUND LENGTH OF SHELL MODE ALLOWABLE PRESSURE BS5500 CODE

412.894 t&C. 2 7.993 t&C. 7.500 ~a 10.000 t&C.

392.893 t&C. 3 5.367 MM. 7.500 ~a 10.000 t&C.

370.895 t&C. 4 6.075 MM. 7.500 t.Pa 10.000 t&C.

345.871 ..... 5 6.420 MM. 7.500 ~a 10.000 t&C.

319.448 ...... 6 6.588 t&C. 7.500 ~a 10.000 t&C.

•• DATA STORED ON FILE MOOLA.SBD u

27

FAILURE PRESSURE

7,088 ~a

6.197 t.Pa

6.414 ~a

6.521 ~a

6.572 ~a

MODEL B

•• INPUT DATA ••

1 THICKNESS OF PLATING 20.0000 UM.

2 FRAME SPACING fJ,00.0000 UM.

3 WIDTH OF rAYING FLANGE IN CONTACT WITH PLATING 0.0000 t.t.t.

4 THICKNESS OF rAYING FLANGE 0.0000 t.t.t.

5 DEPTH OF' FRAME WEB 150.0000 t.t.t.

6 THICKNESS OF' FRAME WEB 15.0000 t.t.f.

7 WIDTH OF FRAME INNER FLANGE 150.0000 t.t.f.

8 THICKNESS OF ~~E FLANGE 15.0000 t.t.f.

1 RADIUS OF MEAN SURFACE OF SHELL PLATING 4000.00 t.t.f.

2 DISTANCE BETWEEN RIGID ENDS 8000.0 t.t.f.

3 YIELD STRESS OF' !SHELL PLATING 600. MPa

~ YIELD STRESS OF I~E FLANGE 450. MPo

5 POISSONS RATIO

6 YOUNGS MODULUS

0.300.

207000. t.Fa


8 OUT OF CIRCULARirY -1.000 MM.

9 WLTIPL:IER FOR RE:SIDUAL FRAME STRESS 1 .20

RADIUS THICKNESS RATIO o/h 200.00


THICKNESS OF' ENDCAP 30 . 0000 t.t.t.

2 RADIUS OF ENDCAP 4000. 0000 t.t.4.


MAXIMUM DESIGN PRESSURE 1.250 MPo

2 SAFETY FACTOR FOR STRESS COMPARISONS- 1.50 SAFETY FACTOR FOR IINTERFRAME BUCKLING COMPARISONS - 2.50 SAFETY FACTOR FOR OVERALL BUCKLING COMPARISONS - 3.50

EFFECTIVE LENGTH OF' SHELL IS FROM BS5500 TABLES

28

•


PRESSURE SAFETY FACTOR

INTERFRAME COLLAPSE

VON MISES ELASTIC INTERFRAME BUCKLING PRESSURE PM WINDENBURG AND TRILLING 2.222 MPa 1. 7774 CHECK

KENDRICKS MINIMIZED MODIFIED VON MISES ELASTIC INTERFRAME BUCKLING PRESSURE AT WAVE NUMBER- 19 2.209 MPa 1. 7672 CHECK

BS5500 LOWER BOUND COLLAPSE CURVE GIVES PM1/PY-0.68 PC/PY-0. 34 AND COLLAPSE PRESSURE • 1.104 MPa 0.8836 CHECK

••• SHELL YIELDING

HOOP STRESS FOR AN UNSTIFFENED CYLINDER PB 3.000 MPa 2.4000

PRESSURE WHERE CIRCUMFERENTIAL STRESS EQUALS YIELD STRESS IN PLATING P3 - WILSON LINEAR FORMULATION 3.042 MPa 2.4333

PRESSURE WHERE MAX. LONGITUDINAL STRESS IN THE PLATING EQUALS THE YIELD STRESS P7- WILSON 2.876 MPa 2.3010

PRESSURE WHERE MEAN MIDBAY CIRCUMFERENTIAL PLATING STRESS EQUALS YIELD STRESS PS - WILSON 3.256 MPa 2.6047

PRESSURE WHERE MEAN MIDBAY STRESS IN PLATING WITH HENCKY-VON MISES YIELD CRITERION EQUALS YIELD 3.755 MPa 3.0040

... FRAME YIELDING

PRESSURE WHERE CIRCUMFERENTIAL STRESS IN THE STANDING FLANGE EQUALS YIELD PFY- WILSON 3.811 I.Fa 3.0488

••• ~E ACCURATE ITERATED SOLUTION FOR STRESS SALERNO AND PULOS NONLINEAR FORMULATION - SHELL

PRESSURE AT WHICH THE MAXIMUM CIRCUMFERENTIAL STRESS IN THE PLATING EQUALS THE YIELD STRESS P3A 2.736 MPa 2.1886

PRESSURE AT WHICH THE LONGITUDINAL STRESS IN THE IN THE PLATING EQUALS THE YIELD STRESS P7A 2.542 MPa 2.0332

PRESSURE AT WHICH THE CIRCUMFERENTIAL STRESS IN PLATING AT MIOBAY EQUALS THE YIELD STRESS P5A 3.004 MPa 2.4032


PRESSURE WHERE CIRCUMFERENTIAL STRESS IN STANDING FLANGE OF THE FRAME EQUALS YIELD - PFYA 6.569 MPa 5.2553

29

•••BS5500 ENDCAP RESULTS

PRESSURE AT WHICH ENDCAP REACHES YIELD

CRITICAL BUCKLING PRESSURE OF ENDCAP

ULTIMATE COLLAPSE PRESSURE FROM BS5500 P/PY CURVE THIS IS THE LOWER BOUND .CURVE OF EXPERIMENTAL DATA

$••OVERALL COLLAPSE

S~ETRIC BUCKLING PRESSURE OF THE RING-SHELL eot.EIINATION - WAVE NUt.I3El~ 0 MODE

BRYANT OVERALL BUCKLING WITH EFFECTIVE WIDTH BRESSE SOLUTION IS FOR A SINGLE STIFFENER AND SHELL SECTION ASSI.lt.t.4ING SHELL FULLY EFFECTIVE

9.000 !.Fa

1-4>.089 !.Fa

2.196 !.Fa

7.823 !.Fa

COLLAPSE PRESSURE MODE EFFECTIVE LENGTH OF SHELL BRESSE BRYANT

2 459.645 0.732 36.306 !.Fa

3 ... 52.826 1.945 7.132 !.Po

... 4#.351 3.628 4.766 !.Fa

5 430.460 5.755 6.086 !.Fa

6 415.269 8.308 8.426 MPo

•••CHECK ON STIFFENER PROPORTIONS FROM BS5500

CRITICAL TRIPPING STRESS OF STIFFENER • THIS IS 0.9390 TIMES YIELD

422.569 !.Po

GENERAL STIFFENER PROPORTIONS 0.00204 LESS THAN CODE RECOMMENDATIONS 0.00217

WEB DEPTH TO THICKNESS RATIO 10.000 IS WITHIN CODE RECOMMENDATIONS 23.592

HALF FLANGE WIDTH TO THICKNESS RATIO 5.000 IS WITHIN CODE RECOMMENDATIONS 10.724

30

7.2000

11.2712

1. 7571

..

6.2580

29.0#7

5.7059

3.8128

4.8685

6.7407


TWO VALUES OF OUT OF CIRCULARITY ARE USED TO DETERMINE THE FAILURE PRESSURE OF THE STIFFENER - KENDRICKS FORMULA WHICH IS A FUNCTION OF WAVE Nl.N3ER AND OVERALL BUCKLING LOAD AND EITHER THE BS5500 VALUE OR A GIVEN OOC VALUE, BOTH OF WHICH ARE CONSTANT WITH WAVE NUt.EER.

EFFECTIVE OUT OF ROUND FAILURE OUT OF ROUND LENGTH OF SHELL ~E ALLOWABLE PRESSURE BS5500 CODE

459.645 ...... 2 538.402 ...... 1.875 ~a 20.000 ......

452.826 ...... 3 29.036 ...... 1.875 ~a 20.000 ......

.......... 351 ...... 4 8.012 ...... 1.875 ~a 20 . 000 w . 430.<460 ...... 5 7.686 ...... 1.875 ~0 20.000 ......

<415.269 t.t.4. 6 8.553 ...... 1 .875 MPa 20.000 ......

•• DATA STORED ON FILE modlb.SBD ••

31

FAILURE PRESSURE

3.093 ~a

2.108 ~a

1.336 ~a

1.263 ~0

1.292 MPa

------------------------------------------------------------------.0---------------·

MODEL G

•• INPUT DATA ••

THICKNESS OF PLATING 23.0000 MM.

2 FRAME SPACING 760.01300 MM.

3 WIDTH OF FAYING FLAN<~E IN CONTACT WITH PLATING 0.0000 MM.

4 THICKNESS OF FAYING I~LANGE 0.0000 MM.

5 DEPTH OF FRAME WEB :!03. 0000 MM.

6 THICKNESS OF FRAME WEE 6.1000 MM.

7 WIDTH OF FRAME INNER FLANGE 101 .0000 t.N.

S THICKNESS OF FRAME FLANGE 8.4000 MM.


2 DISTANCE BETWEEN RIGID ENOS 13680.0 MM.

3 YIELD STRESS OF SHELL PLATING 450. MPa

4 YIELD STRESS OF FRAME FLANGE 450. MPa

5 POISSONS RATIO

6 YOUNGS MODULUS 2071~00. MPa

7 FRAMING TYPE 1. INTEJ~NAL FRAMING

8 OUT OF CIRCULARITY -1 • 000 t.t.4.

9 WLTIPLIER FOR RESIDUAL FRAME STRESS 1.00

RADIL'S THICKNESS RATIO a/h 134.17


1 THICKNESS OF ENDCAP 0.0000 MM.

2 RADIUS OF ENOCAP 0. 0000 MM.



2 SAFETY FACTOR FOR STRESS COMPARISONS- 1.50 SAFETY FACTOR FOR INTERFI~AME BUCKLING Cet.FARISONS - 2.50 SAFETY FACTOR FOR OVERALl.. BUCKLING COMPARISONS - 3.50

EFFECTIVE LENGTH OF SHELL IS FROM BS5500 TABLES

32


PRESSURE SAFETY FACTOR

INTERFRAME COLLAPSE

VON t.USES ELASTIC INTERFRAME BUCI<LING PRESSURE Pt.4 WINOENBURG AAD TRILLING 4.821 t.Fa 2.4107 CHECK

KENDRICKS MINIMIZED MODIFIED VON MISES ELASTIC INTERFRAME BUCKLING PRESSURE AT WAVE NUMBER- 15 4.790 t.Fa 2.3948 CHECK

855500 LOWER BOUND COLLAPSE CURVE GIVES Pt.41/PY•1.37 PC/PY-0.64 AND COLLAPSE PRESSURE • 2.219 t.Fa 1.1096 CHECK

*** SHELL YIELDING

HOOP STRESS FOR AN UNSTIFFENED CYLINDER PB 3.354 MPa 1.6769

PRESSURE WHERE CIRCUMFERENTIAL STRESS EC'UALS YirLD STRESS IN PLATING P3 -WILSON LINEAR FU.iV·vLAilGN 3.3€:9 MFo 1 . t E, ... (.

PRESSURE WHERE MAX. LONGITUDINAL STRESS IN THE PLATING EQUALS THE YIELD STRESS P7 - WILSON 4.198 MPa 2.0990

PRESSURE WHERE MEAN MIDI?..V CIRCUMFERENTIAL PLATING STRESS EQUALS YIELD STRESS PS - WILSON 3.492 t.Fa 1. 7462

PRESSURE WHERE MEAN MIDBAY STRESS IN PLATING WITH HENCKY-VON MISES YIELD CRITERION EQUALS YIELD 4.031 t.Fa 2.0157

••• FRAME YI~LDING

PRESSURE WHERE CIRCUMFERENTIAL STRESS IN THE STANDING FLANGE EQUALS YIELD PFY - WILSON 4.488 t.Fa 2.2438

••• MORE ACCURATE ITERATED SOLUTION FOR STRESS SALERNO AND PULOS NONLINEAR FORWLATION - SHELL

PRESSURE AT WHICH THE MAXIt.U.f CIRCUMFERENTIAL STRESS IN THE PLATING EQUALS THE YIELD STRESS P:SA 3.280 t.Fo 1.6402

PRESSURE AT WHICH THE LONGITUDINAL STRESS IN THE IN THE PLATING EQUALS THE YIELD STRESS P7A 3.934 t.Fo 1.9670

PRESSURE AT WHICH THE CIRCUMFERENTIAL STRESS IN

PLATING AT MIDBAY EQUALS THE YIELD STRESS P5A 3.432 t.Fa 1. 7159


PRESSURE WHERE CIRCUMFERENTIAL STRESS IN STANDING FLANGE OF THE FRAME EQUALS YIELD - PFYA 4.625 t.Fa 2.3126

33

•••OVERALL COLLAPSE

SY),f.AETRIC BUCKLING PRESSURE OF THE RING-SHELL COI.I3INATION - WAVE ~ltR 0 MODE

BRYANT OVERALL BUCKLING WITH EFFECTIVE WIDTH BRESSE SOLUTION IS FOR J1 SINGLE STIFFENER AND SHELL SECTION ASSUMMING SHELL FULLY EFFECTIVE

16.158 t.Fa

COLLAPSE PRESSURE MODE EFFECTIVE LENGTH OF SHELL BRESSE BRYANT

2 429.490 1.452 7.355 MPa

3 418.590 .3.858 4.:!80 MPo

4 406.256 7.200 7.294 MPa

5 388.931 11 .441 11.466 t.Fa

6 370.565 16.555 16.564 t.Fa

•••CHECK ON STIFFENER ~~TIONS FROM BS5500

CRITICAL TRIPPING STRESS OF STIFFENER • THIS IS 0.3598 Tit.tES YII~LO

161.924 t.Fa

GENERAL STIFFENER PROPORriONS LESS THAN CODE RECCMAENOJITIONS

0.00078 0.00217

WEB DEPTH TO THICKNESS RJ,TIO 33.279 GREATER THAN OQDE RECCMAE~DATIONS 23.592

HALF FLANGE WIDTH TO THICKNESS RATIO 6.012 IS WITHIN CODE RECOMMENDATIONS 10.724


TWO VALUES OF OUT OF CIRCULARITY ARE USED TO DETERMINE THE FAILURE PRESSURE OF THE !STIFFENER - KENDRICKS FORt.tULA WHICH IS A FUNCTION OF WAVE NlJ.tBER AND OVERALL BUCKLING LOAD AND EITHER THE 8S5500 VALUE ()R A GIVEN OOC VALUE, BOTH OF WHICH ARE CONSTANT WITH WAVE ~ER.

EFFECTIVE OUT OF ROUND FAILURE OUT OF ROUND LENGTH OF SHELL t.toOE ALLOWABLE PRESSURE BS5500 CODE

429.490 t.tt.t. 2 16.968 t.tt.t. 3.000 t.Fa 15.430 t.tt.t.

418.590 ....... 3 2.022 t.lt.t. 3.000 t.Fa 15.430 t.tt.t.

406.256 ...... 4 3.368 l.t.C. 3.000 t.Fa 15.430 l.t.C.

388.931 l.t.C. 5 4.173 l.t.C. 3.000 MPa 15.430 l.t.C.

370.565 ...... 6 4.612 t.lt.t. 3.000 MPa 15.430 MM.

•• DATA STORED ON FILE modle.SBD ••

34

8.0790

3.6774

2. 18SE c~::cK

3.6468

5.7331

8.2819

FAILURE PRESSURE

3.076 t.Fa

1 .570 t.Fa

1 .646 t.Fa

1. 730 t.Fa

1. 779 M?o

References

1. Gill S.S., 'The Stress Analysis of Pressure Vessels and Pressure Vessel Components', Pergamon Press, Toronto, 1970.

2. BS5500, British Standard Specification for Unfired Fusion Welded Pressure Vessels, Issue 5, April1980.

8. Bushnell D., 'BOSOR4: Program for Stress, Buckling and Vibration of Complex Shells of Revolution', Structural Mechanics Laboratory, Lockheed Missiles and Space Co. Inc., Palo Alto, California.

4. Faulkner D., 'The Collapse Strength and Design of Submarines', RINA Symposium on Naval Submarines, London, 1983.

5. Biezeno C.B., Gramme! R.,'Engineering Dynamics Vol II', Blackie and Sons Ltd., Glasgow, 1956, pp 348-357.

6. European Recommendations for Steel Construction: Buckling of Shells, ECCS - Technical Committee 8, Third Edition, Brussels, 1984.

7. Timoshenko S.P., GereJ.M.,'Theory of Elastic Stability', McGraw-Hill Inc., New York_L 1961.

8. Windenburg D.F ., Trilling C., 'Collapse by Instability of Thin Cylindrical Shells under External Pressure', Trans ASME, Vol 56, 1934.

9. Kendrick S.,'The Buckling Under External Pressure of Ring Stiffened Circular Cylinders', Trans RINA, Vol107, 1965.

10. Salerno V.L., Pulos- J.G.,'Stress Distribution in a Circular Cylindrical Shell Under Hydrostatic Pressure Supported by Equally Spaced Circular Ring Frames - Part 1 Theory', Polytechnic Institute of Brooklyn, June, 1951.

35

------------------------------------------------------------------~----------------------------------~

11. Bryant A.R., 'Hydrostatic Pressure Buckling of a Ring Stiffened Tube', Naval Construction Research Establishment Report 306, October, 1954.

36

1.

3.

'

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ORIGINATOR (the name and 1ddress of the organization preparing the document. 2. DOCUMENT SECURITY CLASSIFICATION Organitations for whom the document was prepared, e.g. Establishment sponsoring (overall security classification of the document. • contractor's report. or tasking agency, are entered in section 12.) including special warning terms if applic3ble)

Defence Research Establishment Atlantic Unclassified

DOCUMENT TITLE !the complete document title as indicated on the title page. Its classification should be indicated by the 1ppropriate abbreviation IS.C.R or Ul in parentheses after the title.)

PRHDEF - STRESS AND STABILITY ANALYSIS OF RING STIFFENED SUBMARINE PRESSURE HULLS

. 4. DESCRIPTIVE NOTES !the category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter the type of

report. e.g. intenm. progress. summary, annual or final. Give the inclusive dates when a specific reporting period is covered.)

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Pegg, N.G. and Smith, D.R.

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JUNE 1987 Annexes. Appendices. etc.) 45 11 ,

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th11 document which the document was written)

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OREA TECHNICAL MEMORANDUM 87 I 213

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! XI Unlimited distribution ( ) Distribution limited to defence departments and defence contractors; further distribution only as approved ( ) Distribution limited to defence departmentS and Canadian defence contractors; further distribution only as approved ( l Distribution limited to government departments and agencies; further distribution only as approved ( l Distribution limited to defence departments; further distribution only IS approved ( ) Other .

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37 SAMPLE PRINT OF FORM OC003

Unclassified SECURITY CLASSIFICATION OF FORM

13. ABSTRACT ( a brief ~nd factual summary of tt.e document. It may also IIPPeat elsewhere in the body of the document itself. It is highly desirable that the abstract of classified document!; be unclassified. E•ch paragraph of the abstract shall begin with ., indication of tho security classification of the information in the pisragraph (unless the document itself is unclassified) represented as (S) (C) (R) or (Ill It is not necessary to include here abS'InCtS in both offlcal languages unless the text is bilingual). ' ' '

Preliminary investigation of the structural integrity of a submarine pressure hull can be accomplished by the use of design formulae. Approximate solutions for stress and stability of uniformly stiffened cylinders subject to hydrostatic pressure have been assembled and incorporated in the computer code PRHDEF •. The British pressure vessel code, BSSSOO, and other codes have been used where appropriate. . Critical pressures are determined for yielding in the frames and shell, for interframE' and overall bifurcation buckling and for collapse of the stiffened shell and endcap. The effect of out of circularity on frame failure is considered and dimension checks for stiffener tripping are made. The background and limitations of the various equations are discussed and results are compared with those obtained using the axisymmetric finite difference program BOSOR4.

The methods described in this report are particularly useful for comparison of various design alternatives on a common basis and for preliminary investigation before more complex and costly finite element or finite difference analyses are undertaken.

1 4. KEVWC?ROS, OE~CRIPTORS or IDENTIFIERS ltechnicall_y meaningful terms or short phrases that characterize a document and could be helpful •n _cata!ogumg the docume:-~ They ~hould be selected so th~ no se.curity classificati~n is required. Identifiers. such as equipment model des•g~at•on, trade name, ·m1htary proJect c1ode name, geographic locat1on may also be mcluded. If possible. keywords should be selected from a published thesaurus. e.g. Thesaurus of Engineering and Scientific Terms (TESn end that thesaurus identified. If it is not possible to select indexing terms which are Unclassified, the classification of each should be indicated as with the title.)

Buckling Stability Pressure hull Submarine Pressure vessel Structural design Ring stiffened

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D R E A

c R D A

S£p 10 1987

Image Cover Sheet CLASSIFICATION SYSTEM NUMBER 52503 … · of uniformly stiffened cylinders subject to hydrostatic pressure have been assembled and incorporated in the comput,er

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